Math Instruction versus Natural Math: Benezet’s Example

Children are intrinsically eager and able to learn. If we step back from our limiting preconceptions about education, we discover learning flourishes when we facilitate it rather than try to advance it through force, intimidation, and coercion.

Over 85 years ago a pioneering educator proved that delaying formal instruction, in this case of mathematics, benefits children in wonderfully unexpected ways. Louis P. Benezet, superintendent of the Manchester, New Hampshire schools, advocated the postponement of systematic instruction in math until after sixth grade. Benezet wrote,

I feel that it is all nonsense to take eight years to get children thru the ordinary arithmetic assignment of the elementary schools. What possible needs has a ten-year-old child for knowledge of long division? The whole subject of arithmetic could be postponed until the seventh year of school, and it could be mastered in two years’ study by any normal child.

While developing this rationale, Benezet spoke with eighth-grade students. He noted they had difficulties putting their ideas into English and could not explain simple mathematical reasoning. This was not only in his district; he found the same results with fourteen-year-old students in Indiana and Wisconsin. Benezet didn’t blame the children or teachers, he blamed introducing formal equations too early. So he began an experiment, abandoning traditional arithmetic instruction below the seventh grade.

In the fall of 1929 I made up my mind to try the experiment of abandoning all formal instruction in arithmetic below the seventh grade and concentrating on teaching the children to read, to reason, and to recite – my new Three R’s. And by reciting I did not mean giving back, verbatim, the words of the teacher or of the textbook. I meant speaking the English language.

To start, he picked out five classrooms, choosing those districts where most students were from immigrant homes and the parents spoke little English. Benezet knew that in other districts the parents with greater English skills and higher education would have vehemently objected, ending the experiment before it started.

In the experimental classrooms, children were exposed to what we’d call naturally occurring math. They learned how to tell time and keep track of the date on the calendar. The students played with toy money, took part in games using numbers, and when dimension terms such as “half” or “double” or “narrower” or “wider” came up incidentally, they were discussed. Instead of math, the emphasis was on language and composition. As Benezet describes these children,

They reported on books that they had read, on incidents which they had seen, on visits that they had made. They told the stories of movies that they had attended and they made up romances on the spur of the moment. It was refreshing to go into one of these rooms. A happy and joyous spirit pervaded them. The children were no longer under the restraint of learning multiplication tables or struggling with long division.

At the end of the first school year, Benezet reported that the contrast between the experimental and traditionally taught students was remarkable. When he visited classrooms to ask children about what they were reading, he described the traditionally taught students as “hesitant, embarrassed and diffident. In one fourth grade I could not find a single child who would admit that he had committed the sin of reading.” Students in the experimental classrooms were eager to talk about what they’d been reading. In those rooms, an hour’s discussion went by with still more children eager to talk.

Benezet hung a reproduction of a well-known painting in the classrooms and asked children to write down anything the art inspired. Another obvious contrast appeared. When he showed the ten best papers from each room to the city’s seventh-grade teachers, they noted that one set of papers showed much greater maturity and command of the language. They observed that the first set of papers had a total of 40 adjectives such as nice, pretty, blue, green, and cold. The second set of papers had 128 adjectives, including magnificent, awe-inspiring, unique, and majestic. When asked to guess which district the papers came from, each teacher assumed that the students who wrote the better papers were from schools where the parents spoke English in the home. In fact, it was the opposite. Those students who wrote the most masterfully were from his experimental classes.

Yet another difference was apparent. It was something that Benezet had anticipated. He explained, “For some years I had noted that the effect of the early introduction of arithmetic had been to dull and almost chloroform the child’s reasoning faculties.” At the end of that first year, he went from classroom to classroom and asked children the same mathematical story problem. The traditionally taught students grabbed at numbers but came up with few correct results, while the experimental students reasoned out correct answers eagerly, despite having minimal exposure to formal math.

Based on these successes, the experiment expanded. By 1932, half of the third- to fifth-grade classes in the city operated under the experimental program. Due to pressure from some school principals, children in the experimental classrooms were back to learning from a math book in the second half of sixth grade. All sixth-grade children were tested. By spring of that year all the classes tested equally. When the final tests were given at the end of the school year, one of the experimental groups led the city. In other words, those children exposed to traditional math curricula for only part of the sixth-grade year had mastered the same skills as those who had spent years on drills, times tables, and exams.

If you have god spatial awareness then your math skills should be alright. I had a colleague who didn’t understand the first thing in math. so she attended one of my classes and said for the first time she actually understood what was going on. And that was a trigonometry class. I think there’s something wrong with the usual methods in schools.

I’ve never understood what trigonometry was FOR. If someone had demonstrated practical applications when I was learning it 40 years ago, I’m sure it would have sunk in a lot better. I do OK with basic arithmetic, and weirdly enough, I used to quite enjoy algebra – it’s all about logic. I completely agree about school methods…

Thank you very much for this informative post. I am not teaching maths formally to my three children, as when I have tried with the older two it has sometimes led to tears and a dislike for maths. It’s interesting how much ‘maths’ comes up in everyday life, and it’s a lot less threatening for the children than formally taught maths. Sometimes I have a confidence wobble though, and it’s good to get a confidence boost.
P.S. In New Zealand we say ‘maths’ instead of ‘math’ 🙂

As with everything, following normal exposure from day-to-day life (which happens in larger amounts when home-schooled), mathematics should only be “taught” to children who want it taught to them. It is much easier and productive to satisfy a curious mind than to force-feed a child who lacks interest.

Thank you for your article. I just recently discovered Benezet myself. I read his three papers last month delineating how he did it. Actually, his 3 Rs were reading, reason, and reciting. He used oral composition in place of written composition in the early years. That too played up the natural propensity of children to tell. This reciting he called oral composition and it built up their working vocabulary and ease of expressing themselves. So both math and language arts appear to be based on incidental, real life, and skills naturally acquired that can come from real life. I love this!

Good point Donna. I also think, from the little reading I’ve done about Benezet, that he understood learning intimately because he was a passion-driven learner and innovative thinker himself. People able to see beyond boundaries put up by cultural conditioning are often the ones to break down those boundaries.

My post linked directly to those articles, but I see now the link is broken. This is extremely frustrating because, as you know, primary sources are vital. The link went to the Benezet Centre, with loads of historic and more current archives. I can access a cached version here:

Hopefully the Benezet Centre is just having some temporary problems and the site will be back up.

If not, this is what the cached page looks like. At least you can search out some of the links by title and author, or contact the researchers listed at the bottom of the page.

Benezet Centre

Over 70 years ago in Manchester, New Hampshire, children learnt no formal arithmetic until grade 6 (about age 11). The program’s creator, Superintendent Louis Benezet, describes it like this:

Picture of Benezet at age 59
In the fall of 1929 I made up my mind to try the experiment of abandoning all formal instruction in arithmetic below the seventh grade and concentrating on teaching the children to read, to reason, and to recite – my new Three R’s. And by reciting I did not mean giving back, verbatim, the words of the teacher or of the textbook. I meant speaking the English language. I picked out five rooms – three third grades, one combining the third and fourth grades, and one fifth grade.

The paragraph above is from the first part of his classic three-part paper:
L. P. Benezet, “The Teaching of Arithmetic I, II, III: The Story of an Experiment,” Journal of the National Education Association
Volume 24(8): 241-244 (November 1935) html| pdf
Volume 24(9): 301-303 (December 1935) html| pdf
Volume 25(1): 7-8 (January 1936) html| pdf
The articles were reprinted in the Humanistic Mathematics Newsletter #6: 2-14 (May 1991).
Here is a single PDF or html file with all three parts. Here are scanned-in GIF images of the articles (as reprinted in the Humanistic Mathematics Newsletter).

Here are JPEG images of Louis P. Benezet.

Related articles

Rote learning

Here is problem 5 from Benezet’s list of problems:

The distance from Boston to Portland by water is 120 miles. Three steamers leave Boston, simultaneously, for Portland. One makes the trip in 10 hours, one in 12, and one in 15. How long will it be before all 3 reach Portland?
In the ninth-grade students in Manchester, traditionaly taught, 6 out of 29 gave the right answer; the experimental second grade “had an almost perfect score.” Probably many of the ninth graders gave 37 as the answer. At least, that is the result of arithmetic teaching today, as these extracts from current research indicates.
Hassler Whitney

Whitney (now deceased), a research mathematician at the Institute for Advanced Study, got interested in elementary education and saw great value in Benezet’s approach.
F. M. Hechinger, “Learning Math by Thinking,” New York Times, 10 June 1986. Reports on the article by Hassler Whitney (the next item in this list).
Hassler Whitney, “Coming Alive in School Math and Beyond,” Journal of Mathematical Behavior, 5(2): 129-40 (Aug 1986). Abstract:
The status of mathematics instruction, especially in the elementary school, is discussed. A meaningful, holistic approach is advocated, rather than an emphasis on rules and procedures.
A. Lax, “Hassler Whitney 1907-1989 – Some recollections 1979-1989,” Humanistic Mathematics Newsletter #4:2-7 (1989).
Andrew Gleason of Harvard wrote a short, unpublished article (html |pdf) arguing that we should try Benezet’s experiment today. (The article is based on a talk he gave at the University of Illinois.) We agree, and hope that people will consider its relevance to other subjects. Physics teaching, for example, suffers from the same rote learning that Benezet abolished in his mathematics reform. What would a Benezet-style physics or science curriculum look like? What about history or languages?
D. Hammer, “Physics for first-graders?” To appear in Science Education (“Comments and Criticism”). Preprint available (html | pdf). Very interesting article about rote-learning in physics.
John Clement, Jack Lochhead, George S. Monk. Translation Difficulties in Learning Mathematics. American Mathematical Monthly 88(4):286-290 (Apr 1981). In PDF: high resolution (574k) or fax resolution (167k).
Mahajan, S. & Hake R.R. 2000. Is It Time for a Science Counterpart of the Benezet-Berman Mathematics Teaching Experiment of the 1930’s? Physics Education Research Conference 2000: Teacher Education.
Material at http://www.whimbey.com. Just as calculators can turn mathematics into button pushing, algebra can turn it into symbol pushing. Solving problems without algebra encourages graphical, visual methods of solution — methods that require thought.
National Research Council, Reshaping School Mathematics: A Philosophy and Framework for Curriculum (Mathematical Sciences Education Board, 1990); pp. 30-31:
Mastery of subject matter has for years been the predominant focus of mathematics education research…Contrary to much present practice, it is generally most effective to engage students in meaningful, complex activities focusing on conceptual issues rather than to establish all building blocks at one level before going on to the next level (Hatano, 1982; Romberg and Carpenter, 1986; Collins et al., 1989)
…
There is some evidence to suggest that paper and pencil calculation involving fractions, decimal long division, and possibly multiplication are introduced far too soon in the present curriculum. Under currently prevalent teaching practice, a very high percentage of high school students worldwide never masters these topics – just what one would expect in a case where routinized skills are blocking semantic learning (e.g., Benezet, 1935). The challenge for curriculum development (and research) is to determine when routinized rules should come first and when they should not, as well as to investigate newer whole-language strategies for teaching that may be more effective than traditional methods. this is an area where more research needs to be done. [Our empahsis.]
Send comments and contributions to:

Sanjoy Mahajan
Richard Hake
We thank:

Jerry Becker, of Southern Illinois University, for sending the plain text of Benezet’s and Gleason’s articles.
Alvin White, of Harvey Mudd College, for reprinting Benezet’s paper in the Humanistic Mathematics Newsletter.
The authors of HTMLDOC, the free-software program used to convert HTML to PDF.

I’m sharing it with my Gifted and Talented group today, as well. WONDERFUL article. I’ve squabble about this for years, especially as it pertains to the horrid experience we give Kindergarteners nowadays.

Very interesting and it rings true to me. And for the handful of kids who are drawn to math at an early age, this method of teaching still opens that door for them.

It seemed to us that a lot of the recommended curriculum in the early grades was unnecessary. Third grade just looked like a repeat of second grade, so for our daughter we skipped it. It all makes you wonder how much of our institutional education is dictated more by traditional or institutional inertia than by its effectiveness.

I feel a lot of the built in redundancy of our current methods in elementary school is due to the natural unfolding of child development. The first reason I believe the repetition is there to catch those who were not developmentally ready the first time. The second reason is that skills not needed in daily life grow rusty. But we repeat the idea that children’s brain slide during the summer. The first reason would be totally unnecessary if formal instruction was given when most children attain that level of brain development. The second reason would be unnecessary if we waited to teach certain concepts until they were more likely to be reinforced by the need in real life.The brain changes at puberty and cognitive development blossoms, abstract learning blossoms at that time too. Math when taught in books is abstract. Those squiggly characters mean what we tell children they mean. Common core tries to rectify it with hands on first. But that still ignores both brain development and practical daily use.

It appears that Bezenet was building on concrete thinking, which is what most children do best in their years before puberty. Taking this natural strength of children he taught them to think and reason with what they learned concretely with math that was practical for them. He builds on children’s ability to tell what they see and know, helping them refine that through reciting. Even their oral expression was based on the concrete experience of what they saw, experienced, and read. Once these powers of reading, reasoning, and reciting were developed and then at puberty, he was able to take advantages of cognitive development and teach the more abstract concepts found in rote learning.

Delayed formal math instruction was nothing new. How he did it was. In the late 1800s there were one year math courses. Children in one room school houses were often taught in one school year what children in city schools spent about eight years to learn. These were usually taught about junior high age. The second year was Euclidean Geometry or Surveying. I own two such math books, one used in Kentucky by my grandfather and his older brothers, and one used in New York by my husband’s grandfather. Country children naturally worked and helped on the farm and lived math in a way city children did not. Most country children learned math by living it, before formal instruction. It appears that Bezenet tried to teach practical math experientially in the classroom and math reasoning, to make up for the lack of experience on the farm.

I love that he actually carried out an experiment and demonstrated the soundness of building on how children naturally develop. He also said in the lower grades his math each day was about 20 minutes. That matches the attention span of children that age.

It is amazing what children can do when we focus on building on their strengths and recognizing where they are developmentally.

Hi, Laura.
I have a B. Sc. in Mathematics and couldn’t agree more. A huge mistake is lying in the current way of teaching Math, yet poor performance seems not to be enough to prompt a change.
Maybe you know the book “Why Johnny Can’t Add: The Failure of the New Math” by Morris Kline.

Curriculum based on Benezet’s approach isn’t possible because it’s learning unique to the child and the situation. Kids learned math in the Benezet experiment as it naturally occurred in their daily lives. That’s possible in the classroom, home, or community today too. If you’re playing a board game, kids end up learn counting and scoring. If you’re making plans, it kids end up learning to tell time and track dates on the calendar. If you’re building something or baking something, kids learn to measure.

Laura lives on a small farm with her family where she works as an editor while also slooowly writing one of the 17 books she alleges she'll actually finish.

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